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Walking on Minimax Paths for -NN Search

Walking on Minimax Paths for -NN Search. Kye-Hyeon Kim and Seungjin Choi Machine Learning Laboratory POSTECH, Korea AAAI-13. Motivation: Clustered Data. Motivation: -NN Search. Query. Motivation: Euclidean Distance. Query. Link-based Measures.

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Walking on Minimax Paths for -NN Search

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  1. Walking on Minimax Pathsfor -NN Search Kye-Hyeon Kim and Seungjin Choi Machine Learning Laboratory POSTECH, Korea AAAI-13

  2. Motivation: Clustered Data

  3. Motivation: -NN Search Query

  4. Motivation: Euclidean Distance Query

  5. Link-based Measures • Pseudo-Inverse of Graph Laplacian(’03, ’07) • (Laplacian) Exponential Diffusion Kernel (’03) • Von Neumann Diffusion Kernel (’04) • Euclidean Commute Time Distance (’04) • Regularized LaplacianKernel (’05) • Markov Diffusion Kernel (’06) • Cross-Entropy Diffusion Matrix (’06) • Random Walk with Restart (’06, ’08) • Regularized Commute Time Kernel (’12) • (See Fouss et al. 2012 and references therein)

  6. Link-based Measures

  7. Neighborhood Graph Connect Close Points

  8. Link-based Measures: Example

  9. Link-based Measures: Example Connect Close Points

  10. Link-based Measures: -NN Graph Time Connect Close Points

  11. Link-based Measures: Example

  12. Link-based Measures: Example More & Shorter Paths  More Similar

  13. Link-based Measures: Example More & Shorter Paths  More Similar

  14. Link-based Measures: Example More & Shorter Paths  More Similar

  15. Link-based Measures: Example More & Shorter Paths  More Similar

  16. Link-based Measures: Example More & Shorter Paths  More Similar

  17. Link-based Measures: Example More & Shorter Paths  More Similar

  18. Link-based Measures: Example More & Shorter Paths  More Similar

  19. Link-based Measures: Example More & Shorter Paths  More Similar

  20. Link-based Measures: Computation Evaluation Needed for All Possible Paths between Nodes More & Shorter Paths  More Similar

  21. Link-based Measures: Computation Similarity = Sum of Scores over All Possible Paths between Nodes More & Shorter Paths  More Similar

  22. Link-based Measures: Computation Similarity = Sum of Scores over All Possible Paths between Nodes More & Shorter Paths  More Similar

  23. Link-based Measures: Computation Similarity = Sum of Scores over All Possible Paths between Nodes More & Shorter Paths  More Similar

  24. Link-based Measures: Computation Similarity = Sum of Scores over All Possible Paths between Nodes More & Shorter Paths  More Similar

  25. Link-based Measures: Computation Similarity = Sum of Scores over All Possible Paths between Nodes Naïve Algorithm: Time Recent Algorithm: Time (Yen, Mantrach, & Shimbo 2008) More & Shorter Paths  More Similar

  26. Shortest Path Distance Shorter Path Exists  More Similar

  27. Shortest Path Distance: Computation Distance = Total Length of Shortest Path between Nodes Dijkstra’s Algorithm: Time Shorter Path Exists  More Similar

  28. Shortest Path Distance: “Shortcuts” Shortcuts Make Distinct Clusters Closer

  29. Problem Summary & Our Goal Poor Scalability of Link-based Measures Poor Robustness of Shortest Path Distance

  30. Problem Summary & Our Goal Poor Scalability of Link-based Measures Poor Robustness of Shortest Path Distance Link-based Measure on More Reliable Paths than Shortest Paths

  31. New Link-based Measure Similarity = Sum of New Scores over All Possible Paths between Nodes

  32. New Link-based Measure Similarity = Sum of New Scores over All Possible Paths between Nodes

  33. New Link-based Measure: Varying Similarity = Sum of New Scores over All Possible Paths between Nodes Long Paths (Large Norms)  Low Scores

  34. New Link-based Measure: Varying Similarity = Sum of New Scores over All Possible Paths between Nodes

  35. New Link-based Measure: Varying Similarity = Sum of New Scores over All Possible Paths between Nodes

  36. New Link-based Measure: Varying Similarity = Sum of New Scores over All Possible Paths between Nodes

  37. New Link-based Measure: Varying Similarity = Sum of New Scores over All Possible Paths between Nodes

  38. New Link-based Measures: Example

  39. New Link-based Measures: Example 8 13 10 12 7 11

  40. New Link-based Measures: Example 8 13 10 12 7 11

  41. New Link-based Measures: Example 8 13 10 12 7 11

  42. New Link-based Measure: Varying With Small , Similarity Sum of Scores over Only a Few Small Paths Between Nodes

  43. New Link-based Measure: Varying With Small , Similarity Sum of Scores over Only a Few Small Paths Between Nodes When , Distance = Norm of Smallest Path Between Nodes

  44. New Link-based Measure: Varying 1 1 1 1 5 1 1 Loose Path 2 2 2 2 2 2 2 2 Compact Path

  45. New Link-based Measure: Varying 1 1 1 1 5 1 1 2 2 2 2 2 2 2 2

  46. New Link-based Measure: Varying

  47. New Link-based Measure: Varying

  48. New Link-based Measure: Varying When Large :Compact Path Small

  49. New Link-based Measures: Example Long Links between Distinct Clusters

  50. New Link-based Measures: Example

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